An application of the splitting-up method for the computation of a neural network representation for the solution for the filtering equations
This work addresses the challenge of solving filtering equations in applications like weather prediction and finance, but it is incremental as it builds on existing PDE-based methods.
The authors tackled the problem of approximating the solution to filtering equations by combining the splitting-up method with a neural network representation to approximate the unnormalized conditional distribution, and they tested it numerically on Kalman and Benes filters.
The filtering equations govern the evolution of the conditional distribution of a signal process given partial, and possibly noisy, observations arriving sequentially in time. Their numerical approximation plays a central role in many real-life applications, including numerical weather prediction, finance and engineering. One of the classical approaches to approximate the solution of the filtering equations is to use a PDE inspired method, called the splitting-up method, initiated by Gyongy, Krylov, LeGland, among other contributors. This method, and other PDE based approaches, have particular applicability for solving low-dimensional problems. In this work we combine this method with a neural network representation. The new methodology is used to produce an approximation of the unnormalised conditional distribution of the signal process. We further develop a recursive normalisation procedure to recover the normalised conditional distribution of the signal process. The new scheme can be iterated over multiple time steps whilst keeping its asymptotic unbiasedness property intact. We test the neural network approximations with numerical approximation results for the Kalman and Benes filter.